| Line 8: | Line 8: | ||
==Relationship between Z-Transform and F.T.== | ==Relationship between Z-Transform and F.T.== | ||
| − | + | <math>X(\omega) = X(e^{j\omega})</math> | |
| − | + | <math>X(z)=X(re^{j\omega})</math> | |
Then <math>X(z) = F(x[n]r^-n)</math> | Then <math>X(z) = F(x[n]r^-n)</math> | ||
| − | + | <math>X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n}</math> | |
Revision as of 14:08, 30 November 2008
Z Transform
Discrete analog of Laplace Transform
$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} $
Where z is a complex variable.
Relationship between Z-Transform and F.T.
$ X(\omega) = X(e^{j\omega}) $
$ X(z)=X(re^{j\omega}) $ Then $ X(z) = F(x[n]r^-n) $
$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n} $
